Seminormal ring

In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy $$x^3 = y^2$$, there is s with $$s^2 = x$$ and $$s^3 = y$$. This definition was given by as a simplification of the original definition of.

A basic example is an integrally closed domain, i.e., a normal ring. For an example which is not normal, one can consider the non-integral ring $$\mathbb{Z}[x, y]/xy$$, or the ring of a nodal curve.

In general, a reduced scheme $$X$$ can be said to be seminormal if every morphism $$Y \to X$$ which induces a homeomorphism of topological spaces, and an isomorphism on all residue fields, is an isomorphism of schemes.

A semigroup is said to be seminormal if its semigroup algebra is seminormal.